# Controller synthesis for switched systems using approximate dynamic programming

#### Abstract

This thesis develops an approximate dynamic programming (ADP) framework for solving optimal control and stabilization problems for a general discrete-time switched nonlinear system. Some important properties of the relaxed value iterations are derived. It is shown that under some mild conditions, the solution generated by the ADP algorithm is exponentially stabilizing and suboptimal. Furthermore, an important connection between the optimal control problem and the exponential stabilization problem is established. It is proved that a switched nonlinear system is exponentially stabilizable if and only if certain finite-horizon approximate value function of a related optimal control problem is an exponentially stabilizing control-Laypunov function (ECLF). This converse ECLF theorem makes the ADP a universal tool for solving the exponential stabilization problem of a general switched nonlinear system. In addition to the general ADP results, the optimal quadratic regulation problem for switched linear systems (DSLQR problem) is studied in details. A new concept, called the relaxed switched Riccati mapping, is introduced to characterize the relaxed value iteration of the DSLQR problem and an efficient algorithm is proposed to compute the iteration and the corresponding suboptimal policy. Furthermore, a stronger converse ECLF theorem is obtained for switched linear systems. It is shown that a switched linear system is exponentially stabilizable if and only if certain finite-horizon approximate value function of a related DSLQR problem is an ECLF. By the DSLQR results, we also know that this ECLF is piecewise quadratic. This result justifies many of the earlier controller synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. An efficient algorithm based on certain suboptimal solution of the DSLQR problem is also proposed, which is guaranteed to yield an exponentially stabilizing policy, whenever the system is exponentially stabilizable.

#### Degree

Ph.D.

#### Advisors

Hu, Purdue University.

#### Subject Area

Electrical engineering|Systems science

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